For the 2D Keller-Segel equation at critical mass 8π with finite second momentum, all solutions converge asymptotically to a renormalized stationary state concentrating at the center of mass on a logarithmic-in-time scale, without symmetry assumptions.
HosonoGlobal existence for the fully parabolic Keller–Segel system with critical mass on the planearXiv preprint arXiv:2602.03768 (2026)
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Global existence of strong solutions for initial mass ≤ 8π in the 2D Patlak-Keller-Segel-Navier-Stokes system, independent of velocity norm.
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Determination of the long-time dynamics for the 2D Keller-Segel equation at critical mass
For the 2D Keller-Segel equation at critical mass 8π with finite second momentum, all solutions converge asymptotically to a renormalized stationary state concentrating at the center of mass on a logarithmic-in-time scale, without symmetry assumptions.
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Critical mass threshold for the 2D Patlak-Keller-Segel-Navier-Stokes system
Global existence of strong solutions for initial mass ≤ 8π in the 2D Patlak-Keller-Segel-Navier-Stokes system, independent of velocity norm.